Chelluri Lecture

Karen SmithUniversity of Michigan
Resolution of singularities

Thursday, April 18, 2019 - 4:30pm
Malott 251

Algebraic varieties are geometric objects defined by polynomials — you have known many examples since high school, where you learned that a circle can be defined by a polynomial equation such as $x^2+y^2=1$. Polynomials can define incredibly complicated shapes, such as a mechanical arm in medical software or Woody's arm in Toy Story, yet they can be easily manipulated by hand or computer. For this reason, algebraic geometry — the study of algebraic varieties and the equations that define them — is a central research area within modern mathematics. It is also one of the oldest and most beautiful.

In general, a variety can have singular points — places where it is pinched or intersects itself. In this talk, we will discuss Hironaka’s famous theorem on resolution of singularities — a technique to “get rid” of the singular points. We introduce a class of singular varieties called rational singularities that are important because they are well-approximated by their resolutions, and explain how one can use “reduction modulo $p$” to characterize them.

Refreshments are provided following the talk at the A.D. White House.